parallel algorithm of navier-stokes model for...

PARALLEL ALGORITHM OF NAVIER-STOKES MODEL FOR MAGNETIC NANOPARTICLES DRUG DELIVERY SYSTEM ON DISTRIBUTED

PARALLEL COMPUTING SYSTEM

SAKINAH BINTI ABDUL HANAN

UNIVERSITI TEKNOLOGI MALAYSIA

PARALLEL ALGORITHM OF NAVIER-STOKES MODEL FOR MAGNETIC NANOPARTICLES DRUG DELIVERY SYSTEM ON DISTRIBUTED

PARALLEL COMPUTING SYSTEM

SAKINAH ABDUL HANAN

A thesis submitted in fulfillment of the

requirements for the award of the degree of

Master of Philosophy

Faculty of Science

Universiti Teknologi Malaysia

JUNE 2017

iii

To

my beloved husband,

mother, father,

and brothers

who have always given me love,

care and cheer

and whose prayers have always been a source o f great inspiration

for me.

May Allah bless you all.

iv

ACKNOWLEDGEMENT

All praises and thanks are due to Allah for making the completion of this thesis

possible. Peace and blessing of Allah be upon Prophet Muhammad (peace be upon

him).

First of all, I would like to express my deepest gratitude and thanks to my

supervisor, PM Dr Norma Alias for their generous contribution in term of knowledge,

expertise and advice that help bring this work to completion. I am especially indebted

to PM Dr Norma Alias, for providing me with many years of continuous coaching,

guidance, motivation and support. Without her understanding, time and patience, I

may not have made it this far. Not to forget she the one who building up my interest

and inspired me to develop confidence in this research.

Next, I am grateful to the individuals who have kindly offered their assistance

in my time of needs. My special thanks go to Nadia Nofri Yeni, Hazidatul Akma, Siti

Hafilah, Masyitah, Maizatul Nadhirah, Nur Hafizah, Akhtar and Imran for their

wonderful deeds in sharing with me their knowledge and skills that are relevant to the

development of this research. I am also thankful for the valuable friendship that I have

obtained from my friends and colleagues who have given me constant advice, help,

and encouragement throughout the years of my studies.

I acknowledge the Senate and the Management of Universiti Teknologi

Malaysia for giving me the opportunity to pursue my Master studies. I am also pleased

to acknowledge the University for sponsoring my studies and provide necessary

facilities for the preparation of this thesis. I acknowledge the Ibnu Sina Institute for

Fundamental Science Studies, UTM, for giving me the permission to use their

facilities.

My special and sincere thanks are for my beloved family. I thanks my parents,

Prof Dr Abdul Hanan Abdullah and Puan Rohana Yusof, and my brothers for their

help, understanding, prayers, and support for always giving me hope and confidence.

Last but not least, I am especially grateful to my supportive and loving husband,

Muhammad Fiqry Aiman Md.Zubadi who has always believed in me and inspired me

to keep on striving despite the challenges along the way. I truly appreciate all of them

sacrifices, patience, and understanding. Without the support and motivation from all

of them, I would not have done this project successfully.

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vi

ABSTRACT

Integrated mathematical Navier-Stokes model for transportation of drug across the blood flow medium by partial differential equations (PDE) with one dimensional (1D) and two dimensional (2D) parabolic type in cylindrical coordinates system are considered. The process of magnetic nanoparticle drug delivery system is made measurable by identifying some parameter such as magnetic nanoparticle targeted delivery, blood flow, momentum transport, density and viscosity on drug release through blood medium, the intensity of magnetic fields, the radius of the capillary and controllability expression to control the concentration of blood. Finite difference method (FDM) with centre difference formula was used to discretization the mathematical model. This research focuses on two types of discretization controlled by weighted parameter 6 = 1 and 6 = - which are implicit (IMP) and Crank Nicolson(CN) schemes respectively. The implementation of several numerical iterative methods such as Alternating Group Explicit (AGE), Red Black Gauss Seidel (RBGS) and Jacobi (JB) method are used to solve the linear system equation (LSE) and is one of the contributions of this research. The sequential algorithm was developed by using C Microsoft Visual Studio 2010 Software. The numerical result was analysed based on execution time, number of iteration, maximum error, root mean square error, and computational complexity. The grid generation process involved fine grained of large sparse matrix by minimizing the size of interval, increasing the dimension of model and level of time steps. Parallel algorithm was proposed for increasing the speedup of computations and reducing computational complexity problem. The parallel algorithms for solution of large sparse systems were design and implemented supported by the distributed parallel computing system (DPCS) containing 8 processors Intel CORE i3 CPUs employing the Parallel Virtual Machine (PVM) software. The parallel performance evaluation (PPE) in term of execution time, speedup, efficiency, effectiveness, temporal performance, granularity, computational complexity and communication cost were analysed for the performance of parallel algorithm. As a conclusion, the thesis proved that the 1D and 2D Navier-Stokes model is able to be parallelized and parallel AGE method is the alternative solution for the large sparse simulation. Based on numerical result and PPE, the parallel algorithm is able to reduce the execution time and computational complexity compared to the sequential algorithm.

vii

ABSTRAK

Navier-Stokes pemodelan matematik bersepadu bagi pengangkutan ubatmelalui aliran darah dengan menggunakan persamaan pembezaan separa (PDE)dengan satu dimensi (1D) dan dua dimensi (2D), berjenis parabola dalam sistemkoordinat silinder dipertimbangkan. Proses nanopartikel magnet melalui sistemperedaran ubat dalam aliran darah diukur dengan mengenal pasti beberapa parameterseperti penghantaran nanopartikel magnet sasaran, aliran darah, pengangkutanmomentum, ketumpatan dan kelikatan bagi perlepasan ubat melalui pengantaraandarah, keamatan medan magnet, jejari kapilari dan juga ungkapan pengawalan yangmengawal kelikatan darah. Kaedah beza terhingga (FDM) dengan formula pembezaantengah telah digunakan untuk mendiskretasikan model matematik tersebut. Kajian initertumpu kepada dua jenis pendiskretan yang dikawal oleh parameter pemberat 9 = 1

1dan 6 = - yang melibatkan skim tersirat (IMP) dan skim Crank Nicolson (CN) secarakhususnya. Perlaksanaan beberapa kaedah berangka seperti Kelas Tak Tersirat Kumpulan Berarah Berselang-seli (AGE), Gauss Seidel Merah Hitam (RBGS) dan Kaedah Jacobi (JB) digunakan untuk menyelesaikan persamaan sistem linear (LSE) dan merupakan salah satu sumbangan dalam kajian ini. Algoritma berjujukan dibangunkan menggunakan perisisan C Microsoft Visual Studio 2010. Keputusan berangka dianalisis berdasarkan masa perlaksanaan, bilangan lelaran, ralat maksima, ralat punca min kuasa dua, dan kekompleksan pengiraan. Proses penjanaan grid yang dihaluskan lagi bagi matrik berskala besar dengan meminimumkan saiz selang ruang, meningkatkan dimensi model dan peringkat paras masa. Algoritma selari dicadangkan untuk meningkatkan kecepatan pengiraan dan mengurangkan masalah kekompleksan pengiraan. Algoritma selari bagi menyelesaikan masalah sistem yang berskala besar dirangkakan dan dilaksanakan serta disokong oleh sistem pengkomputeran selari teragih (DPCS) yang mengandungi 8 pemproses Intel CORE i3 CPUs menggunakan perisian Parallel Virtual Machine (PVM). Penilaian prestasi selari (PPE) berdasarkan masa pelaksanaan, kecepatan, kecekapan, keberkesanan, prestasi sementara, granulariti, kekompleksan pengiraan dan kos komunikasi dianalisis untuk menilai prestasi algoritma selari. Sebagai kesimpulan, kajian ini membuktikan pemodelan Navier-Stokes bagi 1D dan 2D dapat diselarikan dan kaedah selari AGE merupakan penyelesaian alternatif bagi simulasi berskala besar. Berdasarkan keputusan berangka dan PPE, algoritma selari dapat mengurangkan masa pelaksanaan dan kekompleksan pengiraan apabila dibandingkan dengan algoritma berjujukan.

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CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT vi

ABSTRAK vii

TABLE OF CONTENTS viii

LIST OF TABLES xiii

LIST OF FIGURES xvi

LIST OF SYMBOLS xxii

LIST OF ABBREVIATIONS xxiv

LIST OF APPENDICES xxv

1 INTRODUCTION

1.1 Research Background 1

1.2 Problem Statement 5

1.3 Objectives 5

1.4 Scope of Study 6

1.5 Significance of Study 8

1.6 Thesis Outline 8

TABLE OF CONTENTS

2 LITERATURE REVIEW

2.1 Introduction 12

2.2 Cancer and Tumour Growth Problem 13

2.3 Drug Delivery System 14

2.4 Blood as a Medium of Transportation 15

2.5 Magnetic Nanoparticles 16

2.6 Mathematical Modelling on Magnetic Nanoparticles Drug 18

Delivery System

2.6.1 Partial Differential Equation with Parabolic Type 19

2.6.2 Navier-Stoke Model 21

2.6.3 Optimal Control of Navier-Stokes Model 24

2.7 Finite Difference Method 25

2.7.1 Taylor’s Series Expansion 25

2.7.2 Numerical Schemes 28

2.8 Numerical Iterative Methods 31

2.9 Sequential Algorithm 33

2.9.1 Numerical analysis 34

2.9.1.1 Time execution 34

2.9.1.2 Number of iteration 34

2.9.1.3 Convergence and Maximum error 34

2.9.1.4 Stability 35

2.9.1.5 Consistency 36

2.9.1.6 Root mean square error (RMSE) and 36

Accuracy

2.9.1.7 Computational complexity 36

2.10 Parallel Algorithms 37

2.11 Mathematical Software 42

2.12 Parallel Virtual Machines (PVM) 43

2.12.1 An Overview of PVM 43

2.12.2 Distributed Parallel Computing System (DPCS) 45

Platform

2.12.3 Beowulf Cluster 46

2.13 Parallel Performance Evaluation 48

ix

2.13.1 Execution time 49

2.13.2 Speedup 49

2.13.2.1 Amdahl’s law 49

2.13.2.2 Gustafson-Barsis law 50

2.13.3 Efficiency 51

2.13.3 Effectiveness 52

2.13.4 Temporal Performance 52

2.13.5 Granularity 5 3

2.13.6 Communication Cost 54

2.14 Summary of Literature Review 5 5

3 MATHEMATICAL MODELLING OF NAVIER-STOKES

MODEL

3.1 Introduction 57

3.2 Formulation of the Mathematical Navier-Stokes Model on 58

Magnetic Nanoparticles Drug Delivery System

3.2.1 The Governing Navier-Stokes Model 59

3.2.2 Non-Dimensionalization the Navier-Stokes 61

Model

3.2.3 Controllability of the 1D Navier-Stokes Model 62

3.3 Discretization of the Navier-Stokes Model 64

3.3.1 Discretization of 1D Navier-Stokes Model 64

3.3.2 Discretization of 2D Navier- Stokes Model 68

3.4 Linear System Equation (LSE) of Navier-Stokes Model 74

3.4.1 Linear System Equation (LSE) for 1D Navier- 75

Stokes Model

3.4.2 Linear System Equation (LSE) for 2D Navier- 76

Stokes Model

3.5 Properties of Parameter 79

3.5 Conclusion 80

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4 SEQUENTIAL ALGORITHM

4.1 Introduction 81

4.2 Iterative Methods for Solving 1D Navier-Stokes Model 82

4.2.1 Alternating Group Explicit Method 84

4.2.2 Red Black Gauss Seidel Method 93

4.2.3 Jacobi Method 95

4.3 Iterative Methods for Solving 2D Navier-Stokes Model 98

4.3.1 Alternating Group Explicit Method 100

4.3.2 Red Black Gauss Seidel Method 120

4.3.3 Jacobi Method 122

4.4 Result and Discussion 123

4.4.1 Numerical Result for 1D Iterative Method 124

4.4.2 Numerical Result for 2D Iterative Method 127

4.5 Conclusion 131

5 PARALLEL ALGORITHM

5.1 Introduction 133

5.2 Parallelization of 1D Iterative Methods 135

5.2.1 Parallel Alternating Group Explicit Method 145

5.2.2 Parallel Red Black Gauss Seidel Method 147

5.2.3 Parallel Jacobi Method 150

5.3 Parallelization of 2D Iterative Methods 152

5.3.1 Parallel Alternating Group Explicit Method 158

5.3.2 Parallel Red Black Gauss Seidel Method 160

5.3.3 Parallel Jacobi Method 163

5.4 Result and Discussion 165

5.4.1 Parallel Performance Evaluation for 1D 167

Iterative Methods

5.4.2 Parallel Performance Evaluation for 2D 181

Iterative Methods

5.5 Conclusion 195

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6 RESULT AND DISCUSSION


6.2 Visualization and Simulation of Navier-Stokes Model 197

6.2.1 Visualization of Navier-Stokes Model using 197

Comsol Multiphysics

6.2.2 Simulation of Navier-Stokes Model 199

6.3 Discussion on Numerical Result 204

6.4 Discussion on Parallel Performance Evaluation 206

6.5 Conclusion 210

7 CONCLUSION AND FUTURE RESEARCH


7.2 Conclusion of the Study 213

7.3 Suggestion on Future Research 216

REFERENCES 218

xii

Appendices A-C 229

xiii

TABLE NO.

2.1

2.2

3.1

4.1

4.2 (a)

4.2 (b)

4.2 (c)

4.3

4.4

4.5

4.6 (a)

LIST OF TABLES

TITLE

Changing parameter of 6

Classifications of parallel computer architecture

Properties of parameter values

The computational molecules for 1D SAGE, SRBGS,

and SJB

The computational molecules for 2D SAGE

The computational molecules for 2D SRBGS

The computational molecules for 2D SJB

Numerical result for 1D SAGE, 1D SRBGS and 1D SJB

based on execution time, iteration, maximum error and

RMSE for IMP, 0 = 1 and CN 6 = 0.5 with m = 100

Numerical result for 1D SAGE, 1D SRBGS and 1D SJB


RMSE for IMP 6 = 1 and CN 6 = 0.5 with m =

1000000

Computational Complexity of 1D SAGE, 1D SRBGS,

and 1D SJB per iteration

Computational Complexity of 1D SAGE-IMP, 1D

SRBGS-IMP, and 1D SJB-IMP for 0 = 1 with m = 100

and m = 1000000

PAGE

30

46

79

83

99

100

100

124

125

126

126

4.6 (b)

4.7

4.8

4.9

4.10 (a)

4.10 (b)

5.1 (a)

5.1 (b)

5.1 (c)

5.2 (a)

5.2 (b)

5.2 (c)

5.3

5.4

Computational Complexity of 1D SAGE-CN, 1D 127

SRBGS-CN, and 1D SJB-CN for 6 = 0.5 with m = 100

and m = 1000000

Numerical result for 2D SAGE, 2D SRBGS and 2D SJB 128


RMSE for IMP 6 = 1 and CN 6 = 0.5 with m x m =

10 x 10

Numerical result for 2D SAGE, 2D SRBGS and 2D SJB 128


RMSE for IMP 6 = 1 and CN 6 = 0.5 with m x m =

1000 x 1000

Computational Complexity of 2D SAGE, 2D SRBGS, 130

and 2D SJB per iteration

Computational Complexity of 2D SAGE-IMP, 2D 130

SRBGS-IMP, and 2D SJB-IMP for 6 = 1 with m x

m = 10 x 10 and m x m = 1000 x 1000

Computational Complexity of 2D SAGE-CN, 2D 130

SRBGS-CN, and 2D SJB-CN for 6 = 0.5 with m x

m = 10 x 10 and m x m = 1000 x 1000

The domain decomposition 1D molecule of PAGE 139

method.

The domain decomposition 1D molecule of PRBGS 140

method.

The domain decomposition 1D molecule of PJB method. 141

The domain decomposition 2D molecule of PAGE 154

method

The domain decomposition 2D molecule of PRBGS 155

method

The domain decomposition 2D molecule of PJB method 156

Computational Complexity of 1D PAGE, 1D PRBGS 168

and 1D PJB

Communication cost of 1D PAGE, 1D PRBGS and 1D 168

PJB for IMP, 6 = 1 and CN, 6 = 0.5

5.5

5.6

5.7

5.8

5.9

5.10

5.11

5.12

5.13

5.14

Parallel performance evaluations of 1D PAGE-IMP, 1D 169

PRBGS-IMP and 1D PJB-IMP based on execution time,

speedup, efficiency, effectiveness and temporal

performance for IMP scheme 9 = 1

Granularity of 1D PAGE-IMP, 1D PRBGS-IMP and 1D 173

PJB-IMP for IMP scheme 6 = 1

Parallel performance evaluations of 1D PAGE-CN, 1D 175

PRBGS-CN and 1D PJB-CN based on execution time,


performance for CN scheme 0 = 0.5

Granularity of 1D PAGE-CN, 1D PRBGS-CN and 1D 180

PJB-CN for CN scheme 6 = 0.5

Computational Complexity of 2D PAGE, 2D PRBGS 181

and 2D PJB

Communication cost of 2D PAGE, 2D PRBGS and 2D 181

PJB for IMP, 0 = 1 and CN, 6 = 0.5

Parallel performance evaluations of 2D PAGE-IMP, 2D 182

PRBGS-IMP and 2D PJB-IMP based on execution time,


performance for IMP scheme 6 = 1

Granularity of 2D PAGE-IMP, 2D PRBGS-IMP and 2D 187

PJB-IMP for IMP scheme 6 = 1

Parallel performance evaluations of 2D PAGE-CN, 2D 188

PRBGS-CN and 2D PJB-CN based on execution time,


performance for Crank Nicolson scheme 6 = 0.5.

Granularity of 2D PAGE-CN, 2D PRBGS-CN and 2D 194

PJB-CN for Crank Nicolson scheme 6 = 0.5

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xvi

FIGURE NO. TITLE PAGE

1.1 Structure of Nanoparticle 1

1.2 Structure of nanoparticle platforms for drug delivery 2

1.3 The scope of research 7

1.4 An overview of the thesis research 11

2.1 Tumour grows on beginning cancer to diffusion- 13

limited maximal size.

2.2 Magnetic nanoparticles-based drug delivery due to the 18

x direction of magnetization oriented perpendicular to

the direction of blood vessel feeding the target cells.

2.3 Illustrate infinitesimally small 22

2.4 Finite Difference Grid Point Representation 26

2.5 Basic Sequential Algorithm of Numerical Iterative 33

Methods

2.6 Four stages in designing the parallel algorithms 40

2.7 Basic Parallel Algorithm of Numerical Iterative 41

Methods

2.8 Communication procedure for sending and receiving 44

message between adjacent processors

2.9 (a) Schematic diagram Beowulf cluster of PVM by star 47

network topology at the Ibnu Rush Lab, UTM.

2.9 (b) The facilities Beowulf cluster of PVM by star network 48

topology at the Ibnu Rush Lab, UTM.

LIST OF FIGURES

xvii

2.10 Speedup versus number of processors 51

4.1 The sequential algorithms 82

4.2 The Calculation of AGE Method 84

4.3 The algorithm of 1D SAGE 93

4.4 The grid of red and black points 94

4.5 The algorithm of 1D SRBGS 95

4.6 The algorithm of 1D SJB 98

4.7 The algorithm of 2D SAGE 120

4.8 The algorithm of 2D SRBGS 121

4.9 The algorithm of 2D SJB 123

5.1 Communication for master - slave Model for PVM 134

5.2 The parallel algorithms 135

5.3 Regular memory space allocated for 1D problem 136

5.4 (a) Domain decomposition technique for 1D and

(b) Communication between slaves

137

5.4 (c) Domain decomposition of 1D problem with

neighbour subdomain

138

5.5 Data distribution algorithm for 1D 141

5.6 Communication activities for sending and receiving

process by slaves for 1D

142

5.7 Master algorithm for global convergence test for 1D 143

5.8 Slave algorithm for local convergence test for 1D 144

5.9 Data partitioning by number of processors 145

5.10 Communication activities in sending and receiving

points for 1D PAGE

146

5.11 The parallel algorithms of 1D PAGE 146

5.12 1D PRBGS ordering 147

.13 (a) Communication activities in sending and receiving red

points for 1D PRBGS

148

.13 (b) Communication activities in sending and receiving 148

black points for 1D PRBGS

5.14 The parallel algorithms of 1D PRBGS 149

5.15 Communication activities in sending and receiving 150

points for 1D PJB

5.16 The parallel algorithms of 1D PJB 151

5.17 Regular memory space allocated for 2D problem 152

5.18 (a) Domain decomposition technique for 2D and 153

(b) Communication between slaves

5.19 Communication activities for sending and receiving 156

process by slaves for 2D

5.20 Master algorithm for global convergence test for 2D 157

5.21 Slave algorithm for local convergence test for 2D 158


lines for 2D PAGE

5.23 The parallel algorithms of 2D PAGE 160

5.24 (a) Communication activities in sending and receiving red 161

points for 2D PRBGS

5.24 (b) Communication activities in sending and receiving 162

black points for 2D PRBGS

5.25 The parallel algorithms of 2D PRBGS 162


lines for 2D PJB

5.27 The parallel algorithms of 2D PJB 164

5.28 (a) Speedup versus number of processors at different 166

value matrix m for 1D problem

5.28 (b) Speedup versus number of processors at different 166

value matrix m x m for 2D problem

5.29 (a) Time execution versus number of processors for 1D 170

IMP, 0 = 1

5.29 (b) Speedup versus number of processors for 1D IMP, 170

6 = 1

5.29 (c) Efficiency versus number of processors for 1D IMP, 171

6 = 1

xviii

5.29 (d)

5.29 (e)

5.29 (f)

5.30 (a)

5.30 (b)

5.30 (c)

5.30 (d)

5.30 (e)

5.30 (f)

5.31 (a)

5.31 (b)

5.31 (c)

5.31 (d)

5.31 (e)

5.31 (f)

5.32 (a)

Effectiveness versus number of processors for 1D 172

IMP, 0 = 1

Temporal performance versus number of processors 172

1D IMP, for 0 = 1

Granularity versus number of processors for 1D IMP, 174

0 = 1

Time execution versus number of processors for 1D 176

CN, e = 0.5

Speedup versus number of processors for 1D CN, 177

6 = 0.5

Efficiency versus number of processors for1D CN, 177

6 = 0.5

Effectiveness versus number of processors for 1D CN, 178

6 = 0.5


for 1D CN, 6 = 0.5

Granularity versus number of processors for 1D CN, 179

6 = 0.5


IMP, 0 = 1

Speedup versus number of processors for 2D IMP, 183

6 = 1

Efficiency versus number of processors for 2D IMP, 184

6 = 1

Effectiveness versus number of processors for 2D 185

IMP, 6 = 1


for 2D IMP, 6 = 1

Granularity versus number of processors for 2D IMP, 186

6 = 1


CN, 6 = 0.5

5.32 (b) Speedup versus number of processors for 2D CN, 190

6 = 0.5

5.32 (c) Efficiency versus number of processors for 2D CN, 191

6 = 0.5

5.32 (d) Effectiveness versus number of processors for 2D CN, 191

6 = 0.5

5.32 (e) Temporal performance versus number of processors 192

for 2D CN, 6 = 0.5

5.32 (f) Granularity versus number of processors for 2D CN, 193

6 = 0.5

6.1 A diagram shows the operating principle of the drug 198

delivery device. The colour in the droplet represents

drug concentration, with red indicating a higher

concentration and blue indicating zero concentration.

Note that the membrane length is not shown to scale.

6.2 The movement and the total displacement of 198

nanoparticle encapsulated drug at the peak load of (a)

t = 0 sec, (b) t = 0.2 sec, (c) t = 0.4 sec, (d) t = 0.6 sec,

(e) t = 0.8 sec and (f) t = 1.0 sec.

6.3 (a) Velocity profile for blood for different time 200

6.3 (b) Velocity profile for magnetic nanoparticles for 200

different time

6.4 Velocity distribution of magnetic nanoparticles for 201

different distances (y) between the capillary wall and

magnet

6.5 Velocity profile for blood and magnetic nanoparticles 202

with t = 0.4 sec and y = 8 cm

6.6 (a) Velocity profile for blood with t = 0.4 sec and y = 203

8 cm

6.6 (b) Velocity profile for magnetic nanoparticles with t = 203

0.4 sec and y = 8 cm

6.7 The visualization of different function of 204

controllability expression

xx

xxi

6.8 (a) Time execution versus number of processors 207

6.8 (b) Speedup versus number of processors 207

6.8 (c) Efficiency versus number of processors 208

6.8 (d) Effectiveness versus number of processors 209

6.8 (e) Temporal performance versus number of processors 209

6.8 (f) Granularity versus number of processors 210

xxii

u - Velocity of blood (m/s)

v - Velocity of magnetic nanoparticles (m/s)

P - Pressure (Pa)

^ - Viscosity of blood (kg/ms)

Rm - Radius of magnetic nanoparticles (m)

v - Kinematic viscosity

R - Radius of capillary (m)

H - Magnetic intensity

M - Magnetization of particles (A/ m-1)

B - Magnetic induction

X - Magnetic susceptibility of the particle

- Permeability of free space

VM - Volume of MNP (m3)

m - Mass of MNP (kg)

p - Density of blood (kg/m3)

pp - Density of MNP (kg/m3)

N - Number density of suspended nanoparticles

A - Stokes’ coefficient

t - Time (s)

r - Grid in r direction (m)

x - Grid in x direction (m)

LIST OF SYMBOLS

xxiii

z - Grid in z direction (m)

Tp - Execution time

Sp - Speedup

Ep - Efficiency

Fp - Effectiveness

Lp - Temporal performance

G - Granularity

Tcomp - Computational time

Tcomm - Communication time

Tidie - Idle time

Tcommi - Communication cost for minimal data item

xxiv

1D - One Dimension

2D - Two Dimension

AGE - Alternating Group Explicit

DPCS - Distributed Parallel Computing System

FDM - Finite Difference Method

JB - Jacobi

LSE - Linear System Equation

MDCS - MATLAB Distributed Computing Server

MNPs - Magnetic Nanoparticles

MPI - Message Passing Interface

ODE - Ordinary Differential Equation

PAGE - Parallel Alternating Group Explicit

PDE - Partial Differential Equation

PJB - Parallel Jacobi

PRBGS - Parallel Red Black Gauss Seidel

PVM - Parallel Virtual Machines

RBGS - Red Black Gauss Seidel

SAGE - Sequential Alternating Group Explicit

SJB - Sequential Jacobi

SRBGS - Sequential Red Black Gauss Seidel

LIST OF ABBREVIATIONS

xxv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A Some common PVM routines 229

B Table of Literature Review 230

C.1 Discretization of Controllability 1D Navier-Stokes 240

Model

C.2 Linear System Equation (LSE) for Controllability 1D 244

Navier-Stokes Model

CHAPTER 1

INTRODUCTION

1.1 Research Background

Nanotechnology is a general purpose technology because of its important

effects that are related to most of the industries and people. Nanotechnology can be

applied to many areas of research and development such as medicine, manufacturing,

computing, textiles and cosmetics (Dutta, 2015). Nanotechnology is defined as

‘engineering at a very small scale’ which is shown through a nanoparticle range from

1 to 100 nanometers (nm) (Gupta, 2014). In others description, Figure 1.1 shows

structure of nanoparticles with a range of structure size starting from 1 to 100 nm

(Taylor et al, 2013).

Figure 1.1 Structure of Nanoparticle

In nanoparticle manufacturing, a variety of compositions can be achieved as it

may have many practical applications in a variety of areas such as engineering and

medicine. The nanoparticles of drug delivery and related to pharmaceutical

2

development in the context of nanomedicine should be viewed as science and

technology of nanometer scale complex systems. The nanoparticles that are created

for drug delivery purposes are defined as small particles (< 100nm). This definition

includes nanospheres which the drug is absorbed, dissolved or dispersed throughout

the matrix; and nanocapsules in which the drugs are limited to an oily core that

surrounded by the shell-like wall (Kreuter, 2004).

The nanoscale devices allow the chemotherapeutic drug to be discharged into

the blood vessels, spreading out through the tissue and gaining access to the tumour

cells location. Nowadays, drug delivery system is notable for its capabilities when

compared to traditional delivery via bolus injection (Allen and Cullis, 2004). The

structure of nanoparticle platforms for drug delivery is shown in Figure 1.2 (Schoonen

and van Hest, 2014).

The potential of nanoparticles delivery systems provides chances to diversify

the drug delivery approaches or therapeutic options in cancer disease treatment

(Gabathuler, 2010). However, Timchack (2008) said that the drug delivery system is

being designed to ensure an efficient therapy which meant that no harmful side effects

for the body cells and therefore improving the patient life quality. In other words, there

should be minimal damage inflicted to the healthy cells within the body while ensuring

a good recovery progress.

Lifelong diseases such as cancers or tumour growths are common among

people nowadays. Hence, more studies and researches are needed to be carried out so

that the crucial effects of nanoparticles drug delivery systems can be understood more

thoroughly. Drug delivery systems are recommended particularly as the alternatives

times and non-specific recognition

Figure 1.2 Structure of nanoparticle platforms for drug delivery

3

in the form of nanoparticles in order to maintain the effectiveness of the new drugs

that been developed which are more potent and more complex ever before. Magnetic

nanoparticles platform has the ability to let the physicians detect and treat diseases

such as cancer and cardiovascular disease more effectively than before (Baptista et al,

2013).

The unique advantages of an external magnetic field control have achieved the

purpose of targeted delivery because they can be remotely navigated to the intended

site via application of an external magnetic field gradient. Magnetic therapy is widely

used to assist in curing various diseases. The drug that possessed magnetic

nanoparticles properties will be easier to be controlled by external magnetic field and

it has potential therapeutic usage in the cancer cells as well as helps in controlling the

blood pressure in the blood medium (Kumar and Mohammad, 2011). At stationary

position, the transverse magnetic field is applied externally to a moving electrically

conducting fluid where electrical currents are induced in the fluid. The interaction

between these induced currents and the applied magnetic field produces body forces

with a tendency to move slowly along with the movement of blood and bring the drug

to the targeted cells (Sun et al, 2008).

Mathematical models have been developed in order to understand the process

of magnetic nanoparticle drug delivery system. The development of mathematical

model is to predict, design and control the movement of drug delivery through blood

medium controlling by external magnetic fields. Parameters during the process can be

range from very simple to complex in order to upgrade the quality of the system. Some

authors in the area of blood flow feel that blood can be assumed as Newtonion in

nature especially in large blood vessels. In Wang et al. (2014), the flow treatment of

blood has been assumed to be Newtonian fluid and flow as laminar, incompressible,

unsteady and the flow field is simulated by solving the Navier-Stokes equation.

Numerical methods are capable to solve a complex system of partial

differential equation (PDE) which is almost impossible to be solved analytically. The

Finite Element Method (FEM), Finite Volume Method (FVM) and Finite Difference

Methods (FDM) are some alternative methods to solve the PDE (Peiro and Sherwin,

2005). For other application of drug delivery system, the FDM and FEM have been

4

widely used to solve the models (Siepmann, 2008; Palazzo et al, 2005; Dev et al,

2003). However, the FDM scheme is chosen because this method is simple to

formulate a set of discretized equations from the transport differential equations in a

differential manner (Mitchell and Griffiths, 1980). Besides, this method is

straightforward in determining the unknown values. Only a few researchers in

magnetic nanoparticles drug delivery system solved the model using numerical

methods. Thus, due to this reason, the mathematical Navier-Stokes model magnetic

nanoparticles drug delivery system in this research is solved using FDM. Further

details of FDM will be discussed in Chapter 2.

A large scale of system linear of equations is discretized from the FDM for

simulation and visualization. However, one central processing unit (CPU) is not

enough to compute the large computation. Therefore, the parallelization in solving

the large scale of system linear of equations is great importance. The objective is to

speed up the computation and increase the efficiency by using massively parallel

computers. The domain problem is partitioned into subdomain or equal sized tasks.

Then, the tasks are connected to each other local and global communication. Static

mapping strategy is implemented because this research focuses on the distributed

parallel computing architecture. Further detail to describe the parallel algorithm

design methodology is discussed in Chapter 2.

The magnetic nanoparticle drug delivery system to treat the cancer cell

problem is very interesting. Mishra et al (2008) has conducted the application of

magnetic nanoparticle drug delivery system. However, from the existing work by

Mishra et al (2008), this research implement the parallel algorithm of Navier-Stokes

model in magnetic nanoparticle drug delivery system for 1D and 2D model. Thus, this

research which involved a large scale matrix of the discretization model, a large sparse

computational complexity, intensive large-scale parallel computing system and a huge

memory space to support the simulation with a high-speed solution. All the numerical

methods and parallel programming that are used in visualizing and observing the

changing parameters of phase change simulation models run on Linux operating

system by using distributed parallel computing system (DPCS). The parallel

algorithms are programmed in C language while the communication software tool

involves the use of Parallel Virtual Machines (PVM). The parallel performances are

5

analysed in reference to the numerical result and parallel performances evaluation

(PPE).

1.2 Problem Statement

Magnetic nanoparticle drug delivery system is closely related with

biomechanical researchers due to its relationship with cancerous cell or tumour. Some

problem statements will be explored and discussed through this research. Firstly, on

how to model the mathematical modelling and illustrating the visualization of

magnetic nanoparticles drug within its delivery to the specific targeted cell. The

integrated mathematical modelling that uses Navier-Stokes model with continuity and

momentum equations that are related to parameter changes is developed. Thereafter,

the 1D and 2D model with parabolic type equations are solved by the weighted

parameter central FDM in order to obtain the results. The discretized computational is

conducted for 6 = 1 and 6 = -, which represent Implicit (IMP) and Crank Nicolson

(CN) schemes respectively. Dealing with nanoscale phenomena system with large

sparse matrices. By applying some iterative numerical methods, the impact of domain

decomposition techniques, message passing model and grid generation technique can

be stimulated. Furthermore, the implementation of sequential and parallel algorithms

on the PDE is generated by some numerical iterative methods such as Alternating

Group Explicit (AGE), Red Black Gauss Seidel (RBGS) and Jacobi (JB). PPE will be

conducted to measure the superiority of generate FDM schemes in solving the PDEs.

1.3 Objectives

The objectives of the study are:

1. To formulate and visualize the mathematical Navier-Stokes model for

magnetic nanoparticles on the delivery of the drug through blood flow for

cancer cell treatment.

6

2. To discretize the mathematical models using weighted parameter central FDM

involving a number of parameter changes for the large sparse data set.

3. To develop the sequential and parallel algorithms for the Navier-Stokes model

based on three different numerical iterative methods; AGE, RBGS and JB.

4. To analyze the results of 1D and 2D based on the numerical results and PPE

in solving the Navier-Stokes model.

1.4 Scope of Study

In real phenomenon, mathematics provides a broad foundation in each of these

overlapping areas: pure mathematics, applied mathematics, and statistics. This

research will focus on the mathematical modelling for drug delivery via magnetic

nanoparticle system through the blood flow within the cancer cell treatment

application by using PDE. The non Newtonian fluid in blood medium is derived from

the theory that is based on Navier-Stokes model, which emphasizes on the continuity

and momentum equation. The flow scope of the research in Figure 1.3 shows that this

study is focused on the applied mathematics that deals with mathematical modelling

PDE with parabolic type in order to predict the magnetic nanoparticles drug delivery

system. The mathematical modelling involved the continuity and momentum equation

with the initial and boundary conditions are made to be known. Controllability

expression to control the concentration blood is also considered. The 1D and 2D

problems are discretized by using the weighted parameter method. The numerical

iterative methods that are being considered to be used in the comparison are AGE,

RBGS and JB method. The mathematical modelling of the sequential and parallel

algorithms are implemented to solve the problem of drug delivery for the purpose of

dealing with the cancer cell. The approximate solution that uses COMSOL is utilized

for 3D visualization, Matlab for the 1D and 2D visualization and the DPCS with the

communication platform PVM and C programming is applied. The superior iterative

7

method of FDM is being observed and utilized in the solution process of the Navier-

Stokes model.

Figure 1.3 The scope of research.

8

1.5 Significance of Study

The magnetic nanoparticles for the drug delivery model are significant in the

process of developing the alternative numerical simulation for the treatment of the

cancer cell growth. The application of nanoparticles is assumed to be the solution for

early detection of tumour cell growth. The importance of the Navier-Stokes model in

predicting and visualizing the parameters involved to deliver the magnetic

nanoparticles towards the targeted cells. The implementation of the numerical iterative

methods such as AGE, RBGS and JB methods are suitable to solve the 1D and 2D

model. Besides, the simulation of large sparse matrix for the multidimensional models

on DPCS will help reduce the execution time and increases the performance of

speedup. In addition, the approximation result uses parallel computing which is a fast

prediction of the movement of the magnetic nanoparticles through the blood medium.

This medical practice is related to the evolution of biological systems in cancer cells

treatment. Furthermore, this research is beneficial to the cancer patients and doctors

who are working on the cancer diagnostics and cancer treatments through efficient

stimulation and coordination.

1.6 Thesis Outline

The contents of this research can be divided into seven chapters which includes

an introduction, literature review, mathematical modelling, implementation of the

sequential algorithm, implementation of the parallel algorithm, result, discussion, and

conclusion. Chapter 1 starts with a brief discussion on the introduction, problem

formulating, research objectives, the scope of research, significance of study and the

thesis outlines.

Chapter 2 focused on the review regarding the problem, solution of problem,

methodology, computing platform and analysis. The discussion regarding the

problems includes the cancer related issues, the growth of tumours and magnetic

nanoparticles. The mathematical Navier-Stokes model on the magnetic nanoparticle

9

drug delivery is portrayed as the solution to the problem. The mathematical modelling

is used to deal with PDE with parabolic type. Some common methods that are used to

solve this mathematical modelling are the formulation of FDM and weighted average

parameter. The classical and advanced iterative methods are also being highlighted.

Besides, the computational solution of the sequential and parallel algorithm with the

use of common mathematical software and DPCS are also being studied. This chapter

also includes the importance and purpose of mathematical modelling in predicting the

magnetic nanoparticle drug delivery within the cancer cell treatment. Some important

terms that are related to numerical analysis and PPE such as convergence, consistency,

stability, speedup, efficiency, effectiveness, temporal performance and granularity are

also discussed in the chapter.

The scope in Chapter 3 is focused on the integrated mathematical modelling

that is based on the theory of Navier-Stokes model in the fluid dynamic for blood flow

which are the mass (continuity) and momentum conservation equation for 1D and 2D

model. The flow is expected to take place under the influence of externally applied

magnetic field in the axial direction. The equation of continuity is integrated into each

other together with the equation of momentum equation. The governing equations for

1D and 2D are written in the cylindrical coordinate system (r, z, 0). Discretization by

weighted parameter with the usage of the central FDM as well as LSE is also being

discussed in this chapter. Finally, the presence of controllability expression is injected

to 1D Navier-Stokes model due to control of the concentration of blood in capillary

that influenced the magnetic nanoparticle drug delivery system will be developed.

The contribution of Chapter 4 is the development of sequential algorithms of

the magnetic nanoparticle drug delivery model for 1D and 2D. The continuity and

momentum equation will be solved using some numerical iterative methods. The

numerical iterative methods mentioned are AGE (1D SAGE and 2D SAGE), RBGS

(1D SRBGS and 2D SRBGS) and JB (1D SJB and 2D SJB) method. These numerical

method are compared according to the execution times, number of iterations,

maximum errors and root mean square errors (RMSE).

Chapter 5 convert the iterative methods are being presented in Chapter 4 into

the parallel algorithms which aims to improve the time execution when dealing with

10

large sparse matrix and nanoscale problem. The parallelization of the

multidimensional equation will use the same numerical methods discussed in Chapter

4. The parallel algorithm is implemented on PVM with distributed memory within the

message passing environment. The numerical methods are parallelized into 1D PAGE,

2D PAGE, 1D PRBGS, 2D PRBGS, 1D PJB and 2D PJB. The performances are

measured based on speedup, efficiency, effectiveness, temporal performance and

granularity.

The numerical results and the PPE obtained from Chapter 4 and Chapter 5 are

then discussed in Chapter 6. The visualization of the mathematical model is simulated

by using Comsol Multiphysic in 3D simulation and Matlab R2013a stimulation for 1D

and 2D mathematical model. To validate the results obtained, a comparison of the

velocity of the blood and velocity of drug magnetic nanoparticle is made with Mishra

et al. (2008) as a limiting case where it involved 1D model. Simulation of

controllability expression in 1D mathematical modelling is simulated by using Matlab

R2013a. The mathematical modelling which are related to Navier-Stokes magnetic

nanoparticle drug delivery model of 1D and 2D model are analysed based on the

numerical analysis and PPE by comparing between two type of schemes and three

types of numerical iterative method.

Lastly, Chapter 7 draws on the conclusion of the thesis. Contributions are

highlighted and further studies are suggested. An overview of the thesis research is

described in Figure 1.4.

MAS

TER

THES

IS

11

MATHEMATICAL NAVIER-STOKES

MODEL FOR MAGNETIC

NANOPARTICLES DRUG DELIVERY

SYSTEM THROUGH

BLOOD FLOW FOR CANCER

CELL TREATMENT

CONTROLABILITY MATHEMATICAL NAVIER-STOKES

MODEL FOR MAGNETIC

NANOPARTICLES DRUG DELIVERY

SYSTEM THROUGH BLOOD

FLOW FOR CANCER CELL TREATMENT

SEQUENTIALALGORITHM

SJB

1D SRBGS

SAGE

SJB

2D SRBGS

SAGE

VISUALIZATION AND SIMULATION

PARALLELALGORITHM

PJB

1D PRBGS

PAGE

PJB

2D PRBGS

PAGE

VISUALIZATION AND SIMULATION

Figure 1.4 An overview of the thesis research

218

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